Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups

Abstract

Suppose that $p$ is a prime and $X$ is a finite group with a strongly $p$-embedded subgroup, for example a rank $1$ group of Lie type in characteristic $p$. Let $m$ denote the $p$-rank of $X$ and assume that $m\ge 2$. We say that a faithful $\FF_p X$-module is $k$-active if some element of order $p$ in $X$ acts with exactly $k$ non-trivial Jordan blocks. In this paper, we determine the non-trivial composition factors of the faithful $\FF_pX$-modules which are $k$-active for some $k\le m$.